Joint Frequency Offset and Channel Estimation for OFDM

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1 Joint Frequency Offset and Channel Estimation for OFD Xiaoqiang a, Hisashi Kobayashi, and Stuart C. Schwartz Dept. of Electrical Engineering, Princeton University Princeton, New Jersey {xma, hisashi, Abstract We investigate the problem of joint frequency offset and channel estimation for OFD systems. The complexity of the joint maximum likelihood (L) estimation procedure motivates us to propose an adaptive LE algorithm which iterates between estimating the frequency offset and the channel parameters. Pilot tones are used to obtain the initial estimates and then a decision-directed technique provides an effective estimation technique. The joint modified (averaged) Cramer-Rao lower bounds (CRB) of the channel coefficients and frequency offset estimates are derived and discussed. It is shown that, for the case of a large number of subcarriers in the OFD system, thereisapproximatelya6dblossinthefrequencyoffsetestimate lower bound due to the lack of knowledge of the channel impulse response (CIR). The degradation of the CIR lower bound is less severe and depends on the channel delay spread. We show both analytically and by simulation, that the channel estimate accuracy is less sensitive to unknown frequency offset than the frequency offset estimation is affected by the unknown CIR. Comprehensive simulations have been carried out to validate the effectiveness of the adaptive joint estimation algorithm. I. INTRODUCTION Orthogonal frequency division multiplexing [1] is inherently robust against frequency selective fading, since each subchannel occupies a relatively narrow band, where the channel frequency characteristic is nearly flat. It has already been used in European digital audio broadcasting (DAB), digital video broadcasting (DVB) systems, high performance radio local area network (HIPERLAN) and a wireless local area networks (WLAN). It has been demonstrated that OFD is an effective way of increasing data rates and simplifying equalization in wireless communications [1]. Although the carrier frequency is known to the receiver, a frequency drift is not always non-negligible. Another source of frequency offset is the Doppler shift caused by the relative speed between the corresponding transmitter and receiver or the motion of other objects around transceivers. In some cases this deviation is too large for reliable OFD data transmission. There are two problems affected by the frequency offset: one is the decrease in the amplitude of each sampled value, the other is the introduction of inter-carrier interference (ICI). Both will degrade the performance of an OFD system in This work has been supported, in part, by grants from the New Jersey Center Wireless Telecommunications (NJCWT), the National Science Foundation (NSF) and itsubishi Electric Research Labs, urray Hill, NJ and icrosoft Fellowship Program. terms of bit error rate (BER). Consequently, the frequency offset must be estimated and compensated for at the receiver to achieve high-quality transmission. In addition, it is not possible to make reliable data decisions unless a good channel estimate is available for coherent demodulation. The Doppler shift causes the channel characteristic to change from time to time. The relative speed between OFD transceivers may also change from time to time in an outdoor wide-area wireless environment, which causes the frequency offset to vary. The time-varying nature of both the frequency offset and the CIR requires the need for real-time estimation of both. A number of channel estimation algorithms [3]-[5] and frequency offset estimation algorithms [6]-[8] have been proposed in the literature. Usually, perfect frequency synchronization is assumed in deriving channel estimation algorithms. On the other hand, perfect channel estimation or simply, an additive Gaussian channel model is assumed in deriving the frequency offset. As far as we are aware, there have few studies that address such combined estimation problem. In [7], channel estimation is carried out after the frequency offset is compensated and it deals with only a specific frame structure of IEEE a. Li and Ritcy [8] present a simplified L estimation algorithm of the frequency offset using only demodulated decisions, and their algorithm does not incorporate channel estimation into consideration. The main objective of our study is to investigate the use of L algorithms for joint estimation of the channel frequency offset and the CIR in an OFD system that is subject to slow time varying frequency selective fading. II. SYSTE ODEL AND ASSUPTIONS The schematic diagram of Figure 1 is a baseband equivalent representation of an OFD system. The input binary data is first fed into a serial to parallel (S/P) converter. Each data stream then modulates the corresponding subcarrier by PSK or QA. odulations can vary from one subcarrier to another in order to achieve the maximum capacity or the minimum bit error rate (BER) under various constraints. In this paper we use, for simplicity, only QPSK in all the subcarriers, and to denote the number of subcarriers in the OFD system. The modulated data symbols, represented by complex variables X(0),, X( 1), are then transformed by the inverse fast Fourier transform (IFFT). The output symbols are GLOBECO /03/$ IEEE

2 denoted as x(0),,x( 1). In order to avoid inter-frame interference (IFI 1 ), cyclic prefix (CP) symbols, which replicate the end part of the IFFT output symbols, are added in front of each frame. The parallel data are then converted back to a serial data stream before being transmitted over the frequency selective channel. The received data y(0),, y( 1) corrupted by multipath fading and AWGN are converted back to Y (0),,Y( 1) after discarding the prefix, and applying FFT and demodulation. Input bits Output bits S/P P/S Fig. 1. od De- od IFFT Transmitter FFT Receiver Add cyclic prefix Remove cyclic prefix Baseband OFD system model P/S S/P Channel The channel model we adopt in the present paper is a multipath slowly time varying fading channel, which can be described by y(k) = h l x(k l)+n(k), 0 k 1, (1) where h l s (0 l L 1) are independent complex-valued Rayleigh fading random variables, and n k s (0 k 1) are independent complex-valued Gaussian random variables with zero mean and variance σ 2 for both real and imaginary components. L is the length of the CIR. In the presence of channel frequency offset, the above equation becomes [6] kɛ j2π y(k) =e h l x(k l)+n(k), 0 k 1, (2) where ɛ is the channel frequency offset which is normalized by the subcarrier spacing. We assume the frequency acquisition procedure has been completed so that the channel frequency offset is within one half of an interval of the subcarrier spacing, i.e., ɛ 1 2. If the length of the CP is longer than L, there will be no IFI among OFD frames. Thus we need to consider only one OFD frame with subcarriers in analyzing the system performance. The system model and performance can be easily extended to the case of multiple frames. After discarding the cyclic prefix and performing an FFT at the receiver, we can 1 In the literature, the term intersymbol interference (ISI) is used, but we believe inter-frame interference is more appropriate in this paper. obtain the received data frame in the frequency domain: sin πɛ ( 1)ɛ jπ Y (m) = sin πɛ X(m)H(m)e + ICI(m)+N(m), (3) where H(m) is the frequency response of the channel at subcarrier m and the set of the transformed noise variables N(m), 0 m 1 are i.i.d. complex-valued Gaussian variables that have the same distribution as n(k), i.e., with mean zero and variance σ 2 n. The noteworthy term in (3) is the ICI(m), which is given as ICI(m) = 1 1 k=0 n =m k(n m+ɛ) j2π X(n)H(n)e, n m. (4) It is not zero if ɛ 0. Equation (3) shows that the frequency offset degrades the amplitude of the received signal in each suncarrier and introduce inter-carrier interference (ICI). In addition, a common phase shift π ( 1)ɛ is introduced to the received signal. That can be used to estimate the frequency offset, as will be discussed in Section V. In this paper we assume the CIR is constant in each OFD frame and varies from frame to frame according to the fading rate. Furthermore, we assume the system has perfect timing synchronization. Notation: We use the standard notations, e.g., () T denotes the transpose, () denotes the complex conjugate operation, () H denotes the Hermitian, underscore letters stand for column vectors and bold letters stand for matrices. III. JOINT CRB FOR FREQUENCY OFFSET AND CIR In this section we will derive the joint CRB (JCRB) for estimates of the frequency offset and the CIR, assuming the transmitted signals are known. First, we write the system model (2) in vector form as y = 1 ΦW H XW L h + n, (5) where h =[h 0,,h ] T, X =[X(0),,X( 1)] T, y =[y(0),,y( 1)] T, n =[n(0),,n( 1)] T, Φ = [1,e j2π ɛ ( 1)ɛ j2π,,e ] T and H = W L h, W L is a L (i 1)(j 1) j2π submatrix of W with e as the element at the i th row and j th column. We also use the notation X = diag(x), which denotes a matrix with X(m) as its (m, m) entry and zeros elsewhere. The probability density function of y given ɛ, X and h is f(y ɛ, X,h) { 1 = exp 1 (2πσn 2 ) 2σ y 1 n 2 ΦW H XW L h 2}. (6) We define the unknown parameters as θ =[ɛ, h T R,h T I ] T, (7) GLOBECO /03/$ IEEE

3 where h R and h I are the real and imaginary parts of the CIR h. The joint CRLB gives a lower bound for the variance of an unbiased estimate of θ. This is CRLB(θ i )=I 1 (θ) ii, (8) where I(θ) is the Fisher information matrix given by { I(θ) ij = E y log f(y θ,x) } log f(y θ,x) θ i θ j A detailed derivation of the joint CRLB and CRB is given in [2]. Here, we only give the results because of limited space: 3σn 2 CRB(ɛ) = 2π 2 σx 2 ( 1)(2 1), (10) h(l) 2 3σn 2 JCRB(ɛ) = π 2 σx 2 ( 1)( +1), (11) h(l) 2 ) JCRB(h) = σ2 n 3( 1)2 (2L + ( 2. (12) 1) σ 2 X It is easy to find the following relationship between JCRB(ɛ) and CRB(ɛ): JCRB(ɛ) CRB(ɛ) (9) = 2(2 1) ( +1), (13) which approaches 4 when, the number of subcarriers, becomes very large. This implies that there is an approximately 6dB loss of CRB(ɛ) when we do not know the CIR. We also know the CRB of CIR [3] when the frequency offset is zero or precisely known: CRB(h) = 2Lσ2 n σx 2. (14) Therefore, we find the following relationship between the CRB and the joint CRB of the CIR JCRB(h) CRB(h) =1+ 3( 1)2 2L( 2 1), (15) which depends on and the channel delay spread L. As goes to infinity, we have JCRB(h) lim CRB(h) =1+ 3 2L, (16) which means that the larger the channel delay spread, the smaller the relative degradation in the joint CRB of the CIR in the presence of the channel frequency offset. Comparing this result with the 6dB loss of the frequency offset when the channel is known, the degradation in the CIR estimation is much smaller. Thus, we observe that the channel estimation accuracy is less affected by the presence of unknown frequency offset than the frequency offset estimation accuracy is affected by the unknown CIR. What remains to be done is to develop an algorithm that can achieve these joint lower bounds. It is important to note that the joint CRBs are independent of the actual values of frequency offset and CIR. IV. DIRECT JOINT L ESTIATION ALGORITH The joint L estimates of ɛ and h are the values that maximize the probability density function 6, or minimize the distance function D(ɛ, h) D(ɛ, h) = y 1 2 ΦW H XW L h. (17) Thus, the L estimates are [ˆɛ, ĥ] = arg min D(ɛ, h). (18) ɛ,h Taking gradients of D(ɛ, h) with respect to ɛ and h and setting them to zero, we have ɛ D(ɛ, h) = 2 I(y H ΨΦW H XW L h)=0, (19) h D(ɛ, h) = 1 (WL H X H WΦ H y) 1 (WH L X H XW L h) =0. (20) It is by no means straightforward to solve the above two equations to obtain the solution of the joint L estimation problem, because we need to solve a set of equations with 2L +1 unknown parameters. There is obviously no explicit solution for the direct joint minimization which is a nonlinear minimization problem. V. ADAPTIVE JOINT L ESTIATION ALGORITH The above minimization problem is actually a highly nonlinear optimization problem with respect to ɛ and h, which can be solved by a steepest descent algorithm. It contains two steps as stated in the previous section. After finding an initial estimate of ɛ and h in the first step, we carry out the following standard steepest descent procedure ɛ p+1 = ɛ p λ p ɛ D(ɛ p,h p ), (21) h p+1 = h p µ p h D(ɛ p,h p ), (22) where ɛ p and h p are the p th estimates of ɛ and h, λ p and µ p are the step sizes and ɛ D(ɛ p,h p ) and h D(ɛ p,h p ) are the gradients of D(ɛ, h) at ɛ p and h. However, a disadvantage of the steepest descent algorithm is its slow convergence rate. Rather than using the steepest descent algorithm for estimating the CIR in each iteration, we will iterate at each step using a simpler estimation procedure by assuming the frequency offset is known and focusing on the CIR estimation. Then assuming that the CIR is known, we apply the steepest descent algorithm to update the estimate of the frequency offset. This iterative procedure is repeated until convergence. To be more precise, we can obtain h k+1 from a simpler least-squares (LS) estimate as h p+1 = (W H L X H XW L ) 1 W H L X H W(Φ p+1 ) H y, (23) j2π ɛp+1 ( 1)ɛp+1 j2π ] T where Φ p+1 =[1,e,,e The initial estimates of the CIR are obtained by using simple LS algorithm assuming there is no frequency offset, i.e., ɛ =0. GLOBECO /03/$ IEEE

4 This assumption is suitable for the case when the receiver has no knowledge about the exact fractional part of the frequency offset. After obtaining the initial estimates of CIR, we use the time domain (TD) estimation algorithm to obtain the initial estimate of ɛ, assuming the estimates of CIR are perfect. At each time instance index k we compute an estimate ɛ 0 k of ɛ as y(k) kɛ 0 k = 2π angle, 1 k 1. (24) h l x(k l) Then, we combine these 1 initial estimates of ɛ to obtain the actual initial estimate of ɛ as ɛ 0 = 1 2 kɛ 0 k (25) ( 1) k=1 The above joint estimation algorithm is especially desirable when we use OFD preambles or training frames, which do not have particular structure. However, in a practical OFD system only some pilot symbols are inserted in the timefrequency grid. In order to apply the adaptive joint estimation algorithm work to this more practical framework, we need to make some modifications in the above algorithm. We have to replace X by X p which is the p th estimates of the transmitted signal. In particular, Equation (23) becomes h p+1 = (W H L (X p ) H X p W L ) 1 W H L (X p ) H W(Φ p+1 ) H y, and the signal detection procedure is carried out by using simple division and signal mapping (i.e., hard decision) { } X p+1 Y = Hard Decision W L h p+1, (26) where the division is component-wise division of two vectors. The following simulation results verify the effectiveness of the above joint L estimation algorithm. Although the initial estimates of the frequency offset and CIR are very poor, especially when the frequency offset is large, the joint estimation algorithm appears to converge to the correct point for both the frequency offset and the CIR. This is showed in Figure 2 and 3. These two figures also validate our derivation of the joint modified CRLB. Unlike the case when the CIR is known and fixed, the SE of the frequency offset can not achieve the joint modified CRLB. However, the difference is very small. Furthermore, the performance of different frequency offsets is the same. Unlike the frequency offset, the joint modified CRLB of CIR can always be achieved by the algorithm whether or not the channel is slowly changing during the transmission. Figure 4, 5 and 6 show the performance of the SE of frequency offset, SE of CIR and BER of the system, respectively, when only 8 pilot symbols are known for those OFD frames with pilots inserted. The performance degrades, but not that much, especially when the SNR is large. In particular, the SE of the frequency offset has an approximately 2dB loss when SE(ɛ) =. However, the performance degradation of SE(h) depends on the frequency offset itself. A larger frequency offset leads to more degradation of SE(h). The same observation can be made for the BER performance. All the degradation comes from the erroneous detection of the transmitted signals. Note that we simulated an uncoded OFD system. Inclusion of a channel coding scheme should improve the overall performance. Figure 7 shows the mean frequency offset, when the joint L estimation algorithm is adopted. It is clear that the joint L estimates of the frequency offset are unbiased when all the transmitted symbols are known. However, if only some pilot symbols are known, the estimates of the frequency offset become biased when E b /N 0 is small, say less than 14dB. Furthermore, the biased estimated values are always smaller (i.e., negative bias) than the actual frequency offset. VI. CONCLUSION We have considered the problem of joint frequency offset and channel estimation for OFD systems. The joint CRB is derived for this problem. It is shown that there is approximately a 6dB loss in the CRB due to lack of knowledge of the channel. The performance loss of the channel estimation depends on the channel delay spread. Since the joint L estimation is seen to be very complex, an iterative procedure is developed for the joint estimation: a straightforward LS estimate of the CIR is made assuming the frequency offset is known and then the time domain frequency offset estimation algorithm is used assuming the CIR is known. By means of simulation, it is shown that this procedure is effective, converges to the correct points, and comes close to achieving the joint CRB for the frequency offset and the CIR. Furthermore, the frequency offset estimate is unbiased when the E b /N 0 is larger than 14dB. REFERENCES [1] L. J. Cimini, Jr., Analysis and simulation of a digital mobile chaneel using orthogonal frequency division multiplexing, IEEE Transactions on Communications, CO-33, July 1985, pp [2] X. a, H. Kobayashi, and S. Schwartz Joint frequency offset and channel estimation for OFD, Technical Report, Princeton University, June [3] X. a, H. Kobayashi, and S. Schwartz E-based channel estimation for OFD, Technical Report, Princeton University, June [4] Ye (Geoffrey) Li, Leonard J. Cimini, Jr., and Nelson R. Sollenberger, Robust channel estimation for OFD systems with rapid dispersive fading channels, IEEE Trans. on Comm., vol. 46, no. 7, pp , July [5] J.-J. van de Beek, O. Edfors,. Sandell, S. K. Wilson, and P. O. Borjesson, On channel estimation in OFD systems, in Proc. IEEE VTC, vol. 2, pp , [6] X. a, C. Tepedelenlioglu, G. B. Giannakis and S. Bararossa, Nondata-aided frequency-offset and channel estimation for OFD with null subcarriers: Identifiability, algorithms, and performance, IEEE JSAC, vol. 19, no. 12, pp , Dec [7] H. Song, Y. You, J. Paik and Y. Cho, Frequency-offset synchronization and channel estimation for OFD-based transmission, IEEE Comm. Letters, vol. 4, no. 3, pp , ar [8] X. Li and J. A. Ritcey, aximum-likelihood estimation of OFD carrier frequency offset for fading channels, Proc. Thirty-First Asilomar Conference, vol. 1, pp , 1997 [9] F. Gini, R. Reggiannini, and U. engali, The modified Cramer-Rao bound in vector parameter estimation, IEEE Trans. on Comm., vol. 46, pp , Jan GLOBECO /03/$ IEEE

5 SE of frequency offset JCRB(cfo) fixed channel Joint L cfo=0.02 fixed channel fixed channel Joint L cfo=0.2 fixed channel faded channel Joint L cfo=0.02 faded channel faded channel Joint L cfo=0.2 faded channel SE of CIR JCRB(h) Joint L cfo=0.02 Joint L cfo= Fig. 2. ean square error of the joint L estimation of the frequency offset when the transmitted signals are known Fig. 5. ean square error of the joint L estimation of the CIR when only pilot symbols are known JCRB(h) Joint L cfo=0.02 Joint L cfo=0.2 Perfect CIR w/o CFO Joint L cfo=0.02 Joint L cfo=0.2 SE of CIR BER Fig. 3. ean square error of the joint L estimation of the CIR when the transmitted signals are known Fig. 6. Bit error rate of the joint L estimation algorithm when only pilot symbols are known JCRB(cfo) Joint L cfo=0.02 Joint L cfo=0.2 cfo=0.02 with pilots cfo=0.2 with pilots cfo=0.02 known symbols cfo=0.2 known symbols SE of frequency offset ean of frequency offset 10 5 Fig. 4. ean square error of the joint L estimation of the frequency offset when only pilot symbols are known Fig. 7. ean of the frequency offset both for known X and pilot symbols only via the joint L estimation algorithm GLOBECO /03/$ IEEE